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A Progressive Decoupling Algorithm for Minimizing the Difference of Convex and Weakly Convex Functions

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  • Welington de Oliveira

    (Mines Paris, Université PSL, Centre de Mathématiques Appliquées (CMA))

  • João Carlos de Oliveira Souza

    (Federal University of Piauí)

Abstract

Commonly, decomposition and splitting techniques for optimization problems strongly depend on convexity. Implementable splitting methods for nonconvex and nonsmooth optimization problems are scarce and often lack convergence guarantees. Among the few exceptions is the Progressive Decoupling Algorithm (PDA), which has local convergence should convexity be elicitable. In this work, we furnish PDA with a descent test and extend the method to accommodate a broad class of nonsmooth optimization problems with non-elicitable convexity. More precisely, we focus on the problem of minimizing the difference of convex and weakly convex functions over a linear subspace. This framework covers, in particular, a family of stochastic programs with nonconvex recourse and statistical estimation problems for supervised learning.

Suggested Citation

  • Welington de Oliveira & João Carlos de Oliveira Souza, 2025. "A Progressive Decoupling Algorithm for Minimizing the Difference of Convex and Weakly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-24, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02574-4
    DOI: 10.1007/s10957-024-02574-4
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    References listed on IDEAS

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    1. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. R. T. Rockafellar & Roger J.-B. Wets, 1991. "Scenarios and Policy Aggregation in Optimization Under Uncertainty," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 119-147, February.
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